The structure of δ-stable minimal hypersurfaces inRn+1

Abstract

Let Mⁿ (n ≥ 3) be a complete δ (> $\frac{(n-1)^2}{n^2}$)-stable minimal hypersurface in an (n + 1)-dimensional Euclidean space $\mathbb{R}$ⁿ⁺¹. We prove that there are no nontrivial L² harmonic 1-forms on M and the first de Rham's cohomology group with compact support of M is trivial. As corollaries, M has only one end. This implies that if M has finite total curvature, then M is a hyperplane.